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ENERGY CONDITIONS and SCALAR FIELD COSMOLOGY by SHAWN WESTMORELAND B.S., University of Texas, Austin, 2001 M.A., University of T

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ENERGY CONDITIONS and SCALAR FIELD COSMOLOGY by SHAWN WESTMORELAND B.S., University of Texas, Austin, 2001 M.A., University of T ENERGY CONDITIONS AND SCALAR FIELD COSMOLOGY by SHAWN WESTMORELAND B.S., University of Texas, Austin, 2001 M.A., University of Texas, Austin, 2004 Ph.D., Kansas State University, Manhattan, 2010 A REPORT submitted in partial fulfillment of the requirements for the degree MASTER OF SCIENCE Department of Physics College of Arts and Sciences KANSAS STATE UNIVERSITY Manhattan, Kansas 2013 Approved by: Major Professor Bharat Ratra Copyright Shawn Westmoreland 2013 Abstract In this report, we discuss the four standard energy conditions of General Relativity (null, weak, dominant, and strong) and investigate their cosmological consequences. We note that these energy conditions can be compatible with cosmic acceleration provided that a repulsive cosmological constant exists and the acceleration stays within certain bounds. Scalar fields and dark energy, and their relationships to the energy conditions, are also discussed. Special attention is paid to the 1988 Ratra-Peebles scalar field model, which is notable in that it provides a physical self-consistent framework for the phenomenology of dark energy. Appendix B, which is part of joint-research with Anatoly Pavlov, Khaled Saaidi, and Bharat Ratra, reports on the existence of the Ratra-Peebles scalar field tracker solution in a curvature-dominated universe, and discusses the problem of investigating the evolution of long-wavelength inhomogeneities in this solution while taking into account the gravitational back-reaction (in the linear perturbative approximation). Table of Contents List of Figures vi Acknowledgements vii Dedication viii 1 Energy conditions in classical General Relativity1 1.1 Introduction . .1 1.2 The weak energy condition . .5 1.3 The null energy condition . .8 1.4 The dominant energy condition . 12 1.5 The strong energy condition . 16 1.6 Convergence conditions . 18 1.7 The interrelationships between energy conditions . 21 2 Cosmology 24 2.1 Introduction . 24 2.2 How energy conditions constrain the cosmic fluid . 25 2.3 The constituents of the cosmic fluid . 28 2.3.1 Matter . 30 2.3.2 Radiation . 31 2.3.3 The scalar field . 31 2.4 Dark energy . 37 2.5 On the possible equations of state . 38 2.6 The effective cosmological fluid . 40 2.7 The Ratra-Peebles scalar field . 42 2.8 Conclusion . 44 A The Einstein-Hilbert action with boundary terms 50 A.1 Introduction . 50 A.2 The cosmological part . 51 A.3 The Ricci part . 52 A.4 The boundary term . 53 A.5 The matter part . 54 A.6 Conclusion . 54 iv B On the existence and stability of the Ratra-Peebles scalar field tracker solution in a curvature-dominated universe 56 B.1 Introduction . 57 B.2 A special solution . 59 B.3 The attractor property . 61 B.4 Gravitational back-reaction . 65 Bibliography 73 v List of Figures 1.1 Interrelationships between energy conditions and convergence conditions. Ac- cording to the conventions of the present work, the case Λ < 0 corresponds to a repulsive (de Sitter) cosmological constant and the case Λ > 0 corresponds to an attractive (anti-de Sitter) one. 22 2.1 For the toy model of Example 2.3.1, we have plotted a(t) (solid blue), φ(t) (dot-dashed red), and w(t) (dashed orange) all on the same graph with time t running along the horizontal axis. We have set c1 = c2 = 0 and c3 = 1=3. 36 2.2 A list of substances. 39 vi Acknowledgments I would like to express my thanks to Bharat Ratra, for his patient support as my research advisor in the Physics Department at Kansas State University. I wish to spotlight Anatoly Pavlov for very special thanks. The content of Appendix B is part of a joint work that he and I were involved with. I am thankful for his suggestions and useful discussions on certain aspects of this work. I would also like to send my warmest thanks to Larry Weaver, Andrew Ivanov, Omer Farooq, Jianxiong \Jason" Li, Data Mania, Raiya Ebini, Pi-Jung Chang, Louis Crane, Alan Guth, Andy Bennett, Khaled Saaidi, Amit Chakrabarti, Michael J. O'Shea, Mathematica, and especially my loving wife Lei Cao for her endlessly dependable support. vii Dedication To Omer: You expressed how eager you were to read this thesis. Here it is. viii Chapter 1 Energy conditions in classical General Relativity 1.1 Introduction The Einstein field equation describes the relationship between the stress-energy tensor Tµν of matter-fields and the geometrical properties of spacetime. In units where c = G = 1 (our preferred choice of units in the present work), the Einstein field equation reads: 1 R − Rg + Λg = 8πT ; (1.1) µν 2 µν µν µν µν where gµν is the metric tensor, Rµν is the Ricci tensor, R := g Rµν, and Λ is the cosmological constant. In AppendixA, the Einstein field equation is derived using a variational technique. In principle, one can take any metric gµν imaginable (for example, a traversable wormhole discussed below) and | as long as its second partial derivatives exist | plug that metric into the left hand side of (1.1) to produce the stress-energy tensor corresponding to that metric. In this way, exact solutions to (1.1) can easily be constructed, but the stress-energy tensors will not necessarily be physically reasonable. It is therefore useful to impose one or more energy conditions. Energy conditions serve to precisely codify certain ideas about what is physically reasonable. In the present work, we will study four energy conditions that are in standard usage. These are: the weak energy condition (WEC), the null energy condition (NEC), the dominant energy condition (DEC), and the strong energy condition (SEC). 1 Most of the source material for the present chapter comes from Section 4.3 of The Large Scale Structure of Space-time 1 by Hawking and Ellis, and Section 2.1 of A Relativist's Toolkit 2 by Eric Poisson. Example 1.1.1 (a traversable wormhole). The following metric, presented here as a line- element in spherical coordinates, describes a traversable wormhole (cf. Morris and Thorne3); b(r)−1 ds2 = e2γ(r)dt2 − 1 − dr2 − r2 dθ2 + sin2 θd'2 ; (1.2) r where b(r) and γ(r) are twice differentiable functions of r subject to certain restrictions: (1) There is a finite positive radial coordinate r0 where b(r0) = r0. (This is the wormhole throat.) (2)1 − b(r)=r ≥ 0 if r0 ≤ r < rh, where rh is positive and possibly infinite. 0 (3) b (r0) < 1. (4) b(r) ! −Λr3=3 as r ! 1. (5) e2γ(r) ! 1 + Λr2=3 as r ! 1. Items (1) - (3) ensure that the spatial geometry has the shape of a spherically symmetric wormhole, with item (3) ensuring that the two spherical volumes on each side of the worm- hole throat are smoothly joined together (cf. Equation (54) in the paper by Morris and Thorne3). Items (4) and (5) ensure that the metric (1.2) corresponds to de Sitter, anti-de Sitter, or Minkowski spacetime in the asymptotic limit of large r depending on whether Λ < 0, Λ > 0, or Λ = 0, respectively. (According to the sign conventions of the present work, de Sitter spacetime has Λ < 0 and anti-de Sitter spacetime has Λ > 0.) When discussing the topic of traversable wormholes, one usually tries to limit the ac- celerations and tidal forces suffered by travelers who pass through the wormhole. We are here omitting such details but the interested reader is referred to the 1988 paper by Morris and Thorne.3 Morris-Thorne wormholes are asymptotically flat and horizonless, but since 2 we allow the possibility that Λ 6= 0, we should not insist on asymptotic flatness. Moreover, in the case where Λ < 0, one expects a cosmological horizon at r = rh. For a dedicated discussion on spherically symmetric traversable wormholes in de Sitter spacetime, the reader is referred to the 2003 paper by Lemos et al.4 According to the Einstein field equation (1.1), the wormhole (1.2) has a stress-energy tensor Tµν with non-vanishing components given by: e2γ(r) (Λr2 + b0(r)) T = tt 8πr2 Λr3 + r + (2rγ0(r) + 1) (b(r) − r) T = rr 8πr2 (b(r) − r)) 2r2(r − b(r))γ00(r) + (rγ0(r) + 1) [2r(r − b(r))γ0(r) − rb0(r) + b(r)] − 2Λr3 T = θθ 16πr 2 T'' = Tθθ sin θ (1.3) It is often convenient to express the stress-energy tensor in terms of an orthonormal basis 0^ 1^ 2^ 3^ at a base point p. In terms of an orthonormal basis (e0^; e1^; e2^; e3^), or its dual (e ; e ; e ; e ), the metric tensor (locally) has the form of the Minkowski metric, and the stress-energy tensor has the form Tµ^ν^ where T0^0^ is the energy density. For i; j 2 f1; 2; 3g, T^i^j = T^j^i is the ^i; ^j-component of stress: the ^i-component of the force exerted by matter-fields across a unit surface with normal vector e^j. For each i 2 f1; 2; 3g, T^i^i is the normal stress in the (spacelike) e^i direction (normal stress is pressure when T1^1^ = T2^2^ = T3^3^). For i 6= j 2 f1; 2; 3g, T^i^j is called shear stress. The components T0^^i = T^i0^, for i 2 f1; 2; 3g, are (the negatives of) the components of energy flux as measured by an observer at p with 4-velocity e0^ (see Misner, Thorne, and Wheeler5 page 138). Since Tµν is a symmetric second rank tensor, one can always find an orthonormal basis (e0^; e1^; e2^; e3^) at each point, with e0^ future-directed, in which Tµ^ν^ has one of the following mathematically possible forms (cf.
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